Logic of predicates with explicit substitutions - Semantic Scholar
Logic of predicates with explicit substitutions Marek A. Bednarczyk Institute of Computer Science P.A.S., Gdansk
We present a non-commutative linear logic | the logic of predicates with equality and explicit substitutions. Thus, the position of linear logic with respect to the usual logic is given a new explanation.
1 The world according to Girard A recent introduction to linear logic, cf. [13], starts with the following explanation of the position of usual logic with respect to the linear. Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. This paper aims at supporting the same idea. Our justi cation of the claim is, however, quite di erent from the one o ered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, i its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a non-commutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials | the modalities introduced by Girard just to facilitate his embedding. Our construction is also di erent from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formul.
1.1 Linear logic In classical or intuitionistic logics there are several equivalent ways of saying what conjunction or disjunction is. It has been argued by Avron, cf. [2], that logical connectives do not exist outside the context provided by the underlying logic, i.e., the underlying consequence relation. In other words, only after de ning a consequence relation it is possible to talk about connectives associated with it. The connectives, then, are classi ed according to properties they satisfy with respect to this consequence relation. It should come as no surprise that the same operation on formul may play di erent connectives for di erent consequence relations. A simple example of this phenomenon is based on the duality of the notion of the multi-conclusion consequence relation. Thus, an operation which is conjunction with respect to one consequence relation is at the same time a disjunction with respect to the dual consequence relation.
Presentation of logics in terms of invertible rules o ers many technical advantages, see e.g., [8, 9]. Avron, cf. [2], has made of it a dogma saying that the (proof-theoretic) meaning of a logical connective should always be given in terms of an invertible rule. The dogma limits the number of potential de nitions of conjunction, say, to the following two clauses only. ?; A B ` i ?; A; B ` (1) ? ` A & B; i ? ` A; and ? ` B; (2) Clause (1) explains conjunction as a way of putting together two consecutive assumptions in a sequent. The other way is to say that to infer a conjunction means to infer both of the conjuncts separately, see (2). Equivalent, Gentzen-style formulation of conditions (1) and (2) can be found in [2]. Thus, (1) is equivalent to the assumption that the consequence relation is closed under the rules (3). Similarly, (2) is equivalent to the assumption that the consequence relation is closed under the rules (4).
?; A; B ` ? ` A; ? 0 ` B; 0 ?; A B ` ?; ? 0 ` A B; ; 0 ?; A ` ?; B ` ? ` A; ? ` B; ? ` A & B; ?; A & B ` ?; A & B `
(3) (4)
In the presence of the structural rules, i.e., exchange, weakening and contraction, conditions (1) and (2) are equivalent. As a result, the two conjunctions are logically equivalent, i.e., the following holds. A B a` A & B (5) Hence, any occurrence of A B in a formula can always be replaced by A & B , and vice versa, without changing the meaning of that formula. The presence of the structural rules is essential. As soon as one of them is dropped the equivalence (5) breaks down. Linear logic considered by Girard in [10, 13] admits exchange as the only structural rule. Consequently, conjunction splits in two: multiplicative and additive &. Same story goes for disjunction which splits into multiplicative # and additive .
1.2 Exponentials The elimination of contraction and weakening seriously limits the expressive power of linear logic. Without some extra provisos it would be impossible to represent in it either classical or intuitionistic logic. Thus, to increase the expressiveness, Girard introduced two exponentials: ! (Of course! ) and ? (Why not? ), together with the following rules.
?; A ` (dereliction ) ? ` (weakening ) ?; !A ` ?; !A ` ?; !A; !A ` (contraction ) !? ` A; ? (ofcourse ) !? `!A; ? ?; !A `
Above, only the inference rules for ! are given. Those for ? are dual.
Notation !? and ? in (of course) rule is used to express the side condition that all the assumptions and conclusions have the form !C , and ?D, respectively. The following logical equivalences hold in linear logic. !A !B a` !(A & B ) and ?A # ?B a` ?(A B ) (6)
1.3 Representing classical logics into linear logic The addition of exponentials helps | classical and intuitionistic logics can now be represented in linear logic. The exposition given below is based on [9]. Let 1 ; : : : ; k ` 1 ; : : : ; ` (7) be a classical sequent. Its translation into linear logic treats in a di erent way formul on the left and on the right. Thus, one de nes two translations on the level of formul: n( ) and p( ). n(a) = a p(a) = a for atomic a n(: ) = p( )? p(: ) = n( )? n( ^ ) = n( ) & n( ) p( ^ ) = ?p( ) & ?p( ) n( _ ) = !n( ) !n( ) p( _ ) = p( ) p( ) n( ) = !(p( )? ) !n( ) p( ) = n( )? p( ) Then, the linear counterpart of (7) is de ned as follows. !n( 1 ); : : : ; !n( k ) ` ?p( 1 ); : : : ; ?p( ` ) (8) In this way classical logic is embedded in linear logic as the following result shows. Proposition 1 Lemma 3.2 of [9]. Provability of a sequent in classical logic is equivalent to the provability of its translation in linear logic.
1.4 Objections against Girard's translation Girard's explanation of the relationship between the usual and linear logics raises several objections.
What is the linear counterpart of the classical conjunction and disjunction? The translation is schizophrenic when it comes to answer the above question. Indeed, two di erent answers are given. { The translation of formul uses the additives, plus exponentials when needed. { On the level of sequents the multiplicatives are used, plus exponentials again. Thus, two classically equivalent sequents: ; 0 ` ; 0 and ^ 0 ` _ 0 are translated into !n( ); !n( 0 ) ` ?p( ); ?p( 0 ) and ! (n( ) & n( 0 )) ` ? (p( ) p( 0 )), respectively. The validity of Proposition 1 crucially depends on (6). The r^ole of the exponentials is not quite clear. The introduction of exponentials adds another level of complexity to logical system that has already been made quite complicated after splitting nitary conjunctions and disjunctions into multiplicative and additive versions. Are they really needed? Moreover, one cannot say that !A and ?A capture the classical content of a linear formula A. This could be demonstrated even in intuitionistic case where only one exponential is present. Namely, given !A and !B it need not be the case
that !A & !B is logically equivalent to a formula of the form !C again. Thus, formul of the form !A do not constitute the classical sublogic of linear logic. The inadequacy of the explanation of the r^ole played by the exponentials in linear logic is also felt by others. For example Galier, cf. [9], suggested a variation obtained by adding the following rules. !A; !B; ? ` ? `?A; ?B; !A & !B; ? ` ? `?A ?B; The rule on the left gives !A & !B a` !(A & B ). This solves one of the problems mentioned above. Exponentials are non-universal. That major ow of the construction has already been stressed by Girard. The point is that the choice of rules governing the exponentials does not de ne them up to logical equivalence. That is, given two pairs of exponentials !1 ; ?1 and !2 ; ?2 it is impossible to prove, e.g., !1 A a` !2 A. Our goal is to demonstrate that another explanation of the connection between the classical and the linear logics is also possible. We o er a translation that does not resort to exponentials at all.
2 Logic of Predicates with Equality
Logicians, see for example [9, 13], often like to simplify the presentation of logics by rejecting negation as a primitive connective. Instead, it is assumed that the atomic formulas are split evenly into positive and negative. Then negation is de ned as (meta)operation on formul with the help of de Morgan rules. Similarly, one also considers implication as a de ned operation if multiplicative conjunction and disjunction are at hand. We resort to this trick not for convenience sake, but out of real necessity. The ban on negation and derived operations, like classical implication, is crucial for our development to go through.
2.1 A sequent-style presentation
The syntax of logic of predicates LP= | a fragment of the classical logic of predicates with equality considered here | is given by the following grammar. ::= a (atoms) j tt (truth) j ^ (conjunction) j ff (falsity) j _ (disjunction) Thus, we consider quanti er free fragment of classical logic. The structure of atoms as predicates and the way in which they allow to discuss negation is explained later in greater detail. Let us turn to a sequent-style presentation of the logic LP=. Let ', , , etc., be metavariables ranging over the formul, and let , , etc., range over sequences of formul of LP= . A sequent system K for the logic is given in Table 1 in the appendix. In the sequel `K means that the sequent ` can be derived in K. In K conjunctions (nullary and binary) and disjunctions are presented as additive conjunction and disjunction, respectively. It is well-known that in the presence of the structural rules the same e ect is obtained by taking the rules for multiplicatives. The preference given to the additive rules facilitates the process of translation of LP= to linear logic. There, the classical conjunction and disjunction are mapped to the additive conjunction and disjunction, respectively.
2.2 Predicates and admissible negation Let e be a vector of k expressions, e = e1 ; : : : ; ek . Each predicate symbol R of
arity k generates two kinds of atomic formul: { R?+(e), an armative atom, states that the predicate R does hold on e. { R (e), a refutative atom, states that the predicate R does not hold on e. Notation R? (e) is used to denote either R+ (e) or R? (e). Now, negation can be seen as a de nable operation. :R+ (e) = R? (e) :R? (e) = R+ (e) :( ^ ) = (: ) _ (: ) :tt = ff :( _ ) = (: ) ^ (: ) :ff = tt It follows easily from the de nition that :(:') ', where ' means the syntactic equality of formul. Thus, negation is an involution. However, the structural and logical rules in Table 1 do not capture the idea that :' is indeed a negation of '. One way of enforcing this is to accept the following axioms. (`:) ` a; :a (:`) a; :a `
The rst says that assuming a and :a leads to contradiction; the second asserts that either a or :a always holds. It follows by induction that '; :' ` and ` '; :' hold for all '. The implication connective is also meta-de nable: ' ) =^ :' _ .
2.3 Equality Equality predicate plays a very special role in mathematics and in computer science. From the perspective of the intended embedding it presence is, simply, indispensable. Our axiomatization is equivalent to the well-established tradition, cf. [15, 8]. (=) ` e = e (=s) e = e0 ; '[e=x] ` '[e0 =x] The rst axiom schema asserts transitivity of equality. The second axiom schema relates substitution to equality. It captures the idea that equals may be substituted for equals. Formally, (=s) says that under assumption that terms e and e0 are equal one can replace some occurrences of e in a formula '[e=x] by e0 . Here, '[e=x] denotes the formula obtained as a result of syntactic substitution of e for all (free) occurrences of x in '. Thus, just like negation, substitution is presented as a part of the metatheory of LP= .
2.4 Theories
The logic of predicates with equality, i.e., its consequence relation, is generated by the set K of rules together with axiom schemas (`:), (:`), (=) and (=s). Any consequence relation which is obtained in such a way is called a theory. In case of classical logic an equivalent de nition is obtained by saying that a theory is determined by xing a proof system like K and a set of formul, viz. the `axioms'
of the theory. The reason is that, in the presence of implication, validity of any sequent is equivalent to validity of a sequent of the form ` . However, as we shall see in section 3, not all logics allow such simpli cation. That is why a more general notion of an axiom is accepted in this paper. Axioms are used also in many other situations. Typically, they are a convenient way of formalizing properties of the domain of objects under consideration.
3 Logic of Predicates with Explicit Substitutions
Substitution is normally considered, just like in section 2, as a part of the metatheory of a logic. This applies not only to logics, but to -calcul and type theories as well. It has been recently realized that more ecient implementations of functional languages can be achieved if one better controls the process of performing a substitution. This calls for frameworks with substitution as a primitive operation. Indeed, a variety of -calcul with explicit substitutions have already been considered, cf [1, 14]. All of them are 2-sorted | the old syntactic class of terms is retained while a new class of substitutions is added. The logic of predicates has already two sorts: the sort of terms and, built over terms, the sort of formul. So far nobody has considered adding explicit substitutions to it via a new syntactic sort. All attempts known to the author use the idea that substitutions behave as modal operators, see e.g., [16]. We have good reasons to consider substitutions as a new kind of atomic formul, cf. [5]. Predicates are eternal. They represent facts the truth of which does not depend on the context, or state, in which their truth is evaluated. Therefore, we call them Platonic here. Substitutions provide a formalization of the idea of an action or change of state in logic. Hence, we call them dynamic atoms. Since a substitution is an atom, a new logical connective is needed to express (pending) application of to, say, a predicate a. Let us use the tensor symbol
to denote the postulated connective. Thus, the above situation could be written as a. Consequently, assuming is associative, (0 a) a` ( 0 ) a. Hence, 0 would corresponds to composition of substitutions. Now, the question is this: Does such a logical connective exist? It turns out that we can view as a non-commutative multiplicative conjunction. The identity substitutions xx , for all x, should be neutral elements of . It is therefore natural to identify them all with I | the multiplicative truth constant. Since I captures the idea that \nothing changes" it is a good speci cation for a \do nothing" program, cf. [5]. Altogether, we are led to discover that the logic of predicates with explicit substitution, called LP= , is a fragment of intuitionistic non-commutative linear logic. Formul of LP= are given by the following grammar. A ::= j a j A A j I j A & A j > j A A j ? Unlike Girard we use ? instead of 0 to denote the neutral element of . This is to stress that it is the least element w.r.t. the derivability relation, and a dual to >. 3.1 A sequent system for LP= A sequent-style presentation of the logic is given in Table 2 in the appendix.
With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [10, 11, 12], to the non-commutative case, cf [7]. The exceptional axiom is (?). Its expected generalisation is ?; ?; ` A, as in [7]. However, the stronger axiom is not valid in our intended interpretation in quasi quantales as described in section 4 and in. [4]. Embedding the usual logic into LP= gives a good reason for not assuming that having falsehood as one of the assumptions always logically implies anything.
3.2 Platonic Formul
The idea underlying our embedding is that the logic of predicates lives as a Platonic sublanguage of LP= . More precisely, a formula is called Platonic if it contains neither explicit substitutions, nor nor I . To put the same statement in positive form, a Platonic formula is built from Platonic atoms and from additive conjunctions and disjunctions. Clearly, the Platonic formul are in 1{1 correspondence with formul of the predicate logic considered in section 2. Thus, with a slight abuse of notation, we let ' range over Platonic foruml of LP= .
3.3 Platonic axioms
Bear in mind that in a classical sequent ` sequences and stand for the additive conjunctions and disjunction of formul in and , respectively. Thus, axioms (:`) and (`:) responsible for negation correspond to (&:) R? (e) & :R? (e) ` ? and (:) > ` R? (e) :R? (e)
The linear counterpart of the re exivity of equality axiom schema is the following. (=) > ` e = e The other facet of equality, that equals can be substituted for equals, re ects the dynamic nature of substitution. This is treated in subsection 3.5. In rst order logic the conjunction distributes over disjunction, i.e., (A B ) & C a` (A & C ) (B & C ) and A (B & C ) a` (A B ) & (A C ): In linear logic not all of the above follow from the rules of Table 2. Consequently, we add those missing as axioms. (&) A (B & C ) a (A B ) & (A C ) (&) (A B ) & C ` (A & C ) (B & C )
3.4 Dynamic atoms
Substitutions, ranged over by , have the form xe .
3.5 Dynamic axioms
The additive connectives and predicate atoms are the Platonic ingredients of LP= . The dynamic ingredients are: the substitutions, multiplicative conjunction, and multiplicative truth. The dynamic features of the logic are captured by the following set of axioms. We add them all to the theory we build.
Substitution versus equality The principal property that equals may be substituted for equals can now be expressed as the following axiom. (=) e1 = e2 & ex1 ` ex2 Quick comparison with its predecessor in the logic of predicates reveals that the formula ' which played a dummy there is now, simply, removed. Pending substitutions Idea: B represents pending upon B.
Substitution pending on a Platonic atom is explained via meta-substitution. (R? ) e ? x R (e1 ; : : : ; ek ) a` R? (e1 [e=x] ; : : : ; ek [e=x]) One substitution pending upon another corresponds to their composition. Thus, composing two substitutions which concern the same variable results in aggregating the e ects of both in a single substitution. Just as in (R? ) pending explicit substitution in the world of formul is reduced/explained by metalevel substitution in the world of terms. All identity substitutions are equal I . (xx) (I ) x e1 e2 a` e2 [e1 =x] x a` I
x
x
x
Since simultaneous substitutions are not allowed, all we can do to explain the e ect of composition of two substitutions for di erent variables is to say how they commute. Again, the equality predicate is needed to state the axioms. (xz ) c1 = e2 [e1 =x] & e1 = c2 [c1 =z ] & ex1 ez2 ` cz1 cx2
Axiom (xz ) has the side-condition that x and z are di erent variables. The meaning of B in other cases is guided by distributivity axioms in subsection 3.5.
Platonic formul are facts The meaning of A B where A is a Platonic formul has not been described yet. Predicates, i.e., the Platonic atoms, were described as facts, i.e., formul the truth of which does not depend on the state. The same property holds for ?, and is enforced on the additive truth. The following axiom schemas capture the idea. (R? ) R? (e ; : : : ; e ) ` R?(e ; : : : ; e ) ? (> ) > ` > ? 1
k
1
k
Notice that axiom (> ) does not co-exist with the more general axiom ?; ?; ` A for the additive false. Together they give inconsistency: > ` ?. As a consequence of the above axioms one obtains the following result. Proposition 2. Platonic 's are a constant predicate transformers: a` B. The above proposition provides a technical justi cation of the idea that the Platonic formul do not depend on the state.
Distributivity axioms In the fragment of linear logic described in Table 2 the only distributivity law which is guaranteed is that of over binary additive disjunction . In general, as exempli ed by (R? ) and (> ), does not distribute over nullary additive disjunction ? on the right. But it does for dynamic atoms. (?) ? ` ? Let us recall that the right-sided distributivity ? A ` ? always holds by (?). Distributivity of over additive conjunctions is not guaranteed in general.
But we want it at least for the binary &. ( &) A (B & C ) a (A B ) & (A C ) (& ) (A & B ) C a (A C ) & (B C ) For nullary conjunction we impose left-sided distributivity only for dynamic atoms. (>) > a >
Its general form is not valid, cf. prop. 2. Its right-sided form always holds by (> ). Elimination of substitutions The consequences converse to those above follow from the logical rules, so the above give a method for elimination of substitutions and other non-Platonic ingredients. Consider the following inductive de nition which shows how it can be done when a formula has a form A '.
Lemma 3. For any Platonic ' and any A the formula dA 'e is a well de ned Platonic formula such that A ' a` dA 'e.
4 Translation of Logic of Predicates to LP= The general idea underlying our translation should be clear by now: Logic of predicates lives as the Platonic fragment of a LP= . Formally, given a classical formula ' its translation into LP= , denoted b'c, is de ned by induction on the structure of ' as follows. bR?(e1 ; : : : ; ek )c = R?(e1 ; : : : ; ek ) bttc = > b' ^ c = b'c & b c bff c = ? b' _ c = b'c b c Now, one can show that the function that eliminates tensor de ned in subsection 3.5 captures the performing of substitution. Lemma 4. d b'ce = b'c. Hence, b'c a` b'c in LP= , by Lemma 3. In LP the following holds
` i V
W
^
`
_
(9)
where and denote the conjunction and disjunction of all formul in and , respectively.
Accordingly, let bc denote the sequence obtained from by applying the translation elementwise. Keeping in mind the connection between the classical connectives and the linear connectives, and the equivalence (9), one de nes the translation on the level of sequents as follows.
&
b ` c def = bc `
M
b c:
4.1 Soundness of the translation
The reader should notice that the axioms responsible for negation adopted in LP= are translations of the negation axioms in predicate logic. Same applies to re exivity of equality axiom. This generalises to arbitrary rst order theories since every axiom of predicate logic corresponds to a Platonic axiom of LP= via translation. Somewhat more subtle situation is in the case of the axiom (=s). Its translation is e=e0 & b'[e=x]c ` b'[e0 =x]c. From prop. 2 it follows e=e0 ` e=e0 '. From h that i lemma 4 it follows that b'[e=x]c ` xe b'c and ex b'c ` b'[e0 =x]c. Thus, the task substantially simpli es. Now, it can be accomplished by the following proof. by (=) h i e=e & xe ` ex b'c ` b'c h i e ex b'c e = e & ; b ' c ` x by (&
) ? distributivity h i (e=e b'c) & ( xe b'c) ` (e=e & xe ) b'c (e=e & xe ) b'c ` ex b'c 0
0
0
0
0
0
h
0
0
0
i
(e=e b'c) & ( xe b'c) ` ex b'c 0
0
&
Before we formulate the main result of this section let usLintroduce a bit of notation. Given a set A of LP= axioms let bAc = f bc ` b c j ` 2 Ag denote its translation to LP= . With the above notation we can formulate the following result. Theorem 5. Let ` be a LP theory obtained by extending K with negation and equality axioms, and an arbitrary set of axioms A. Suppose that ` is a LP= theory obtained by extending L and such that the corresponding negation and equality axioms are satis ed together with (&), (&) and (& ) distributivity laws. Finally, suppose that ` admits all axioms in bAc. Then
` implies
&bc `
4.2 Conservativity of the translation
M
b c:
The result converse to Theorem 5 also holds. The proof presented in this section is based on semantical considerations. For the remaining part of this section let ` be a LP theory obtained by extending K with negation and equality axioms, and an arbitrary set of axioms A. Let ` be obtained by extending L with all axioms listed in section 3, i.e., negation, equality, substitutivity, distributivity and, nally, all axioms from the set bAc.
Denotational semantics of LP The notion of a model M for LP theory is given as follows. First, a set D is chosen, and for each function symbol f of arity k a function fM : Dk ! D called f 's interpretation. These data induce expression interpretation function EM [ ] : Exp ! (Var ! D) ! D, where Var and Exp are the syntactic classes of variables and expressions, respectively. Call Val =^ Var ! D the set of valuations of variables in D. For v : Val, x : Var and d : D de ne a new valuation, denoted v[x 7! d], by v[x 7! d]y =^ d whenever x and y are the same variables, and v[x 7! d]y =^ vy otherwise. It is assumed here that, given a particular vocabulary for building expressions, the reader is capable of lling in the details required to give the de nition of EM [ ] . The denotation of the formul of the logic of predicates in M is given by function FM [ ] : Pred ! Pow(Val)
where Pred is the set of LP= formul. Thus, each formula is identi ed with a set of valuations, viz., the valuations that satisfy the formula. This denotation function is determined once an interpretation RM Dk for each k-ary predicate symbol is given. It is assumed, that the interpretation =M D D is the diagonal binary relation on D. Let e =^ e1 ; : : : ; ek . FM [ R?+ (e)]] = fv : Val j (EM [ e1 ] v; : : : ; EM [ ek ] v) 2 RM g FM [ ff ] ) = ; FM [ R (e)]] = fv : Val j (EM [ e1 ] v; : : : ; EM [ ek ] v) 62 RM g FM [ tt] ) = Val FM [ ' _ ] = FM [ '] [ FM [ ] FM [ ' ^ ] = FM [ '] \ FM [ ] We may consider the substitution as an operation ondenotations of formul. Formally, given a set S of valuations and a substitution xe we de ne
e S =^ fv : Val j v[x 7! E [ e] v] 2 S g : x
h i
With the above notation one obtains the Substitution Lemma. Lemma 6 Substitution. FM [ '[e=x]]] = xe FM [ '] : Every model M determines a consequence relation j=M given by
j=M i
\
[
FM [ ] FM [ ] Call M a model of ` if ` j=M , i.e., whenever ` implies j=M . Let M be the set of all models of ` . Then Godel completeness result follows. Theorem 7 Completeness. ` i j=M , for all M 2 M.
Denotational semantics of LP= The notion of a model for a LP= theory=is
obtained by a trivial adaptation of the idea of LP-model. That is, each LP model is characterised by exactly the same set of ingredients as before. We take the same denotations for expressions. All the di erence are focused on the area of denotations of formul. Whereas before these were set of valuations, now they are functions between valuations.
Formally, the denotation of the formula of the logic of predicates with explicit substitutions in model M is given by function SM [ ] : Pred ! (Pow(Val) ! Pow(Val)) where Pred is the set of LP= formul. SM [ R? (e)]]S = FM [ R? (e)]] SM [ xe ] S = xe S SM [ A B ] S = SM [ A] (SM [ B ] S ) SM [ I ] S = S SM [ A B ] S = SM [ A] S [ SM [ B ] S SM [ ?] )S = ; SM [ A & B ] S = SM [ A] S \ SM [ B ] S SM [ >] )S = Val The semantic function extends to sequences in accord to the intuition that the meaning of a sequence stands for iterated tensor. Let us write ? j=M A i SM [ ? ] S SM [ A] S , for all S Val.
Proposition 8. 1. For any LP formula ' its dynamic meaning (via translation) is a constant function given by: SM [ b'c] S = FM [ '] . 2. b'c j=M b c implies ' j=M . It is time to come back to the issue of negation as a `second class citizen' in the logic of predicates. Taking it to the rst class league' would result in many problems. Simply, there can be no negation in our intended dynamic denotational semantics. At the moment we have not de ned how to interpret non-intuitionistic sequents ? ` , where is not a singleton. But, assuming that we have done it, somehow, the problem remains. Negation :A of a linear formula A should satisfy at least :A; A ` ?. But then :I; I ` ?, and hance :I a` ?. Since :> a` ? should also hold the negation could not be involution.
The result First, notice that under assumptions about ` and `, every model of the former is also a model of the latter. Proposition 9. Let M be a model of `. Then it is also a model of `. And now we are ready to formulate the main result.
Theorem 10. ` is a conservative extension of `
&
L
Proof. Let bc ` b c. Without loss of generality we may assume that and consist of a single formul ' and , respectively. Let M be a model of ` . By prop. 9 it is also a model of ` . So, b'c j=M b c. Hence, by prop. 8 one obtains ' j=M . Finally, ' ` , by prop. 7. ut
Conclusions We have shown that it is possible to give an explanation of the connections between the usual logic and linear logics based on the idea that the former is a platonic fragment of a speci c linear theory LP= | the logic of predicates with explicit substitutions.
Admittedly, the target linear logic is rather weak: it is non-commutative, has only the additives and multiplicative conjunctions, and even lacks one of the implications | see [4] for details. The interest in studying LP= stems from the fact that its formul are natural speci cations of imperative programs in a mechanised variant of Hoare logic. Thus, I is a natural and best speci cation for skip. Similarly, xe captures the meaning of the assignment x := e. Finally, sequential composition of programs is mimiced by tensor of their speci cations. See [5] for details.
Acknowledgements
I would like to acknowledge stimulating discussions with colleagues at ICS PAS, and in particular I would like to thank Beata Konikowska and Andrzej Tarlecki. The research presented here was partially sponsored by The State Committee for Scienti c Research, Grant No 2P301 007 04.
References 1. Abadi, M., Cardelli, L., Curien, P.-L. and J-J. Levy. Explicit substitutions. ACM Conf. on Principles of Programming Languages, San Francisco, 1990. 2. Avron, A. Simple consequence relations. Information & Computation, Vol 92, No 1, pp. 105{139, May 1991. 3. Basin, D., Matthews, S., L.Vigano. A modular presentation of modal logics in a logical framework. Proc. 1st Isabelle User Workshop, pp.137{148, Cambridge, 1995. 4. Bednarczyk, M. A. Quasi quantales. GDM Research Report, ICS PAS, 1994. 5. Bednarczyk, M. A. and T. Borzyszkowski Towards program development with Isabelle. Proc. 1st Isabelle User Workshop, pp.101{121, Cambridge, 1995. 6. Bednarczyk, M. A. Logic of predicates versus linear logic. ICS PAS Reports, No 795, December 1995. 7. Brown, C. and D. Gurr. Relations and non-commutative linear logic. To appear in J. of Pure and Applied Algebra 8. Gallier, J. Logic for Computer Science. Foundations of Automatic Theorem Proving. Harper & Row, New York, 1986. 9. Gallier, J. Constructive logics. Part II: linear logic and proof nets. Research Report, CIS, University of Pennsylvania, 1992. 10. Girard, J.-Y. Linear logic. Theoretical Computer Science, 50 (1987), pp 1{102. 11. Girard, J.-Y. and Y. Lafont. Linear logic and lazy computation. In: Proc. TAPSOFT'87 (Pisa), vol. 2, LNCS 250, Springer Verlag, 1987, pp 52{66. 12. Girard, J.-Y., Lafont, Y. and P. Taylor. Proofs and Types. volume 7 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989. 13. Girard, J.-Y. Linear logic : its syntax and semantics. In Advances of Linear Logic, 1995. 14. Lescane, P. From to v a journey through calculi of explicit substitutions. ACM Conf. on Principles of Programming Languages, San Portland, 1994. 15. Mendelson, E. Introduction to Mathematical Logic. D. Van Nostrand Company, 1964. 16. Poigne, A. Basic Category Theory. In Handbook of Logic in Computer Science. Vol I.. Clarendon Press, Oxford, 1992.
Appendix ` ; ' '; ` '`' ; ` ; ` ; ; '; '; ` (` Contr) (Contr `) '; ` ` ; ` ` (` Weak) (Weak `) '; ` ` ; ; '; ; ` ` ; '; ; (Exch `) (` Exch) ; ; '; ` ` ; ; '; (` tt) (ff `) ff ; ` ` ; tt '; ` ; ` ` ; ' ` ; (_`) (`^) ' _ ; ` ` ; ' ^ ` ; ' '; ` (`_ l) (^` l) ' ^ ; ` ` ; ' _ ` ; ; ` (`_ r) (^` r) ' ^ ; ` ` ; ' _ = Table 1. A sequent system K for PL . (Id)
0
0
0
0
0
0
0
A; ` B (Cut) ? ` A; ?; ; `B 0
(Re ) A ` A (LI ) ?;?;I;``AA
0
(L ) ?;?;AA; B;B;``CC A; ` C (L&-L) ?; ?; A & B; ` C
B; ` C (L&-R) ?; ?; A & B; ` C (?) ?; ? ` A A (R-L) ? ?` A` B
0
(Cut)
(RI ) ` I A `B (R ) ? ?;` ` A B (>) ? ` > (R&) ? `?A` A &?B` B
(L) ?; A; ?;`AC B; ?;`B;C ` C
B (R-R) ? ?` A` B Table 2. A sequent system L for NILL.
This article was processed using the LATEX macro package with LLNCS style
We present a non-commutative linear logic | the logic of predicates with equality and explicit substitutions. Thus, the position of linear logic with respect to the usual logic is given a new explanation.
1 The world according to Girard A recent introduction to linear logic, cf. [13], starts with the following explanation of the position of usual logic with respect to the linear. Linear logic is not an alternative logic ; it should rather be seen as an extension of usual logic. This paper aims at supporting the same idea. Our justi cation of the claim is, however, quite di erent from the one o ered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, i its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a non-commutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials | the modalities introduced by Girard just to facilitate his embedding. Our construction is also di erent from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formul.
1.1 Linear logic In classical or intuitionistic logics there are several equivalent ways of saying what conjunction or disjunction is. It has been argued by Avron, cf. [2], that logical connectives do not exist outside the context provided by the underlying logic, i.e., the underlying consequence relation. In other words, only after de ning a consequence relation it is possible to talk about connectives associated with it. The connectives, then, are classi ed according to properties they satisfy with respect to this consequence relation. It should come as no surprise that the same operation on formul may play di erent connectives for di erent consequence relations. A simple example of this phenomenon is based on the duality of the notion of the multi-conclusion consequence relation. Thus, an operation which is conjunction with respect to one consequence relation is at the same time a disjunction with respect to the dual consequence relation.
Presentation of logics in terms of invertible rules o ers many technical advantages, see e.g., [8, 9]. Avron, cf. [2], has made of it a dogma saying that the (proof-theoretic) meaning of a logical connective should always be given in terms of an invertible rule. The dogma limits the number of potential de nitions of conjunction, say, to the following two clauses only. ?; A B ` i ?; A; B ` (1) ? ` A & B; i ? ` A; and ? ` B; (2) Clause (1) explains conjunction as a way of putting together two consecutive assumptions in a sequent. The other way is to say that to infer a conjunction means to infer both of the conjuncts separately, see (2). Equivalent, Gentzen-style formulation of conditions (1) and (2) can be found in [2]. Thus, (1) is equivalent to the assumption that the consequence relation is closed under the rules (3). Similarly, (2) is equivalent to the assumption that the consequence relation is closed under the rules (4).
?; A; B ` ? ` A; ? 0 ` B; 0 ?; A B ` ?; ? 0 ` A B; ; 0 ?; A ` ?; B ` ? ` A; ? ` B; ? ` A & B; ?; A & B ` ?; A & B `
(3) (4)
In the presence of the structural rules, i.e., exchange, weakening and contraction, conditions (1) and (2) are equivalent. As a result, the two conjunctions are logically equivalent, i.e., the following holds. A B a` A & B (5) Hence, any occurrence of A B in a formula can always be replaced by A & B , and vice versa, without changing the meaning of that formula. The presence of the structural rules is essential. As soon as one of them is dropped the equivalence (5) breaks down. Linear logic considered by Girard in [10, 13] admits exchange as the only structural rule. Consequently, conjunction splits in two: multiplicative and additive &. Same story goes for disjunction which splits into multiplicative # and additive .
1.2 Exponentials The elimination of contraction and weakening seriously limits the expressive power of linear logic. Without some extra provisos it would be impossible to represent in it either classical or intuitionistic logic. Thus, to increase the expressiveness, Girard introduced two exponentials: ! (Of course! ) and ? (Why not? ), together with the following rules.
?; A ` (dereliction ) ? ` (weakening ) ?; !A ` ?; !A ` ?; !A; !A ` (contraction ) !? ` A; ? (ofcourse ) !? `!A; ? ?; !A `
Above, only the inference rules for ! are given. Those for ? are dual.
Notation !? and ? in (of course) rule is used to express the side condition that all the assumptions and conclusions have the form !C , and ?D, respectively. The following logical equivalences hold in linear logic. !A !B a` !(A & B ) and ?A # ?B a` ?(A B ) (6)
1.3 Representing classical logics into linear logic The addition of exponentials helps | classical and intuitionistic logics can now be represented in linear logic. The exposition given below is based on [9]. Let 1 ; : : : ; k ` 1 ; : : : ; ` (7) be a classical sequent. Its translation into linear logic treats in a di erent way formul on the left and on the right. Thus, one de nes two translations on the level of formul: n( ) and p( ). n(a) = a p(a) = a for atomic a n(: ) = p( )? p(: ) = n( )? n( ^ ) = n( ) & n( ) p( ^ ) = ?p( ) & ?p( ) n( _ ) = !n( ) !n( ) p( _ ) = p( ) p( ) n( ) = !(p( )? ) !n( ) p( ) = n( )? p( ) Then, the linear counterpart of (7) is de ned as follows. !n( 1 ); : : : ; !n( k ) ` ?p( 1 ); : : : ; ?p( ` ) (8) In this way classical logic is embedded in linear logic as the following result shows. Proposition 1 Lemma 3.2 of [9]. Provability of a sequent in classical logic is equivalent to the provability of its translation in linear logic.
1.4 Objections against Girard's translation Girard's explanation of the relationship between the usual and linear logics raises several objections.
What is the linear counterpart of the classical conjunction and disjunction? The translation is schizophrenic when it comes to answer the above question. Indeed, two di erent answers are given. { The translation of formul uses the additives, plus exponentials when needed. { On the level of sequents the multiplicatives are used, plus exponentials again. Thus, two classically equivalent sequents: ; 0 ` ; 0 and ^ 0 ` _ 0 are translated into !n( ); !n( 0 ) ` ?p( ); ?p( 0 ) and ! (n( ) & n( 0 )) ` ? (p( ) p( 0 )), respectively. The validity of Proposition 1 crucially depends on (6). The r^ole of the exponentials is not quite clear. The introduction of exponentials adds another level of complexity to logical system that has already been made quite complicated after splitting nitary conjunctions and disjunctions into multiplicative and additive versions. Are they really needed? Moreover, one cannot say that !A and ?A capture the classical content of a linear formula A. This could be demonstrated even in intuitionistic case where only one exponential is present. Namely, given !A and !B it need not be the case
that !A & !B is logically equivalent to a formula of the form !C again. Thus, formul of the form !A do not constitute the classical sublogic of linear logic. The inadequacy of the explanation of the r^ole played by the exponentials in linear logic is also felt by others. For example Galier, cf. [9], suggested a variation obtained by adding the following rules. !A; !B; ? ` ? `?A; ?B; !A & !B; ? ` ? `?A ?B; The rule on the left gives !A & !B a` !(A & B ). This solves one of the problems mentioned above. Exponentials are non-universal. That major ow of the construction has already been stressed by Girard. The point is that the choice of rules governing the exponentials does not de ne them up to logical equivalence. That is, given two pairs of exponentials !1 ; ?1 and !2 ; ?2 it is impossible to prove, e.g., !1 A a` !2 A. Our goal is to demonstrate that another explanation of the connection between the classical and the linear logics is also possible. We o er a translation that does not resort to exponentials at all.
2 Logic of Predicates with Equality
Logicians, see for example [9, 13], often like to simplify the presentation of logics by rejecting negation as a primitive connective. Instead, it is assumed that the atomic formulas are split evenly into positive and negative. Then negation is de ned as (meta)operation on formul with the help of de Morgan rules. Similarly, one also considers implication as a de ned operation if multiplicative conjunction and disjunction are at hand. We resort to this trick not for convenience sake, but out of real necessity. The ban on negation and derived operations, like classical implication, is crucial for our development to go through.
2.1 A sequent-style presentation
The syntax of logic of predicates LP= | a fragment of the classical logic of predicates with equality considered here | is given by the following grammar. ::= a (atoms) j tt (truth) j ^ (conjunction) j ff (falsity) j _ (disjunction) Thus, we consider quanti er free fragment of classical logic. The structure of atoms as predicates and the way in which they allow to discuss negation is explained later in greater detail. Let us turn to a sequent-style presentation of the logic LP=. Let ', , , etc., be metavariables ranging over the formul, and let , , etc., range over sequences of formul of LP= . A sequent system K for the logic is given in Table 1 in the appendix. In the sequel `K means that the sequent ` can be derived in K. In K conjunctions (nullary and binary) and disjunctions are presented as additive conjunction and disjunction, respectively. It is well-known that in the presence of the structural rules the same e ect is obtained by taking the rules for multiplicatives. The preference given to the additive rules facilitates the process of translation of LP= to linear logic. There, the classical conjunction and disjunction are mapped to the additive conjunction and disjunction, respectively.
2.2 Predicates and admissible negation Let e be a vector of k expressions, e = e1 ; : : : ; ek . Each predicate symbol R of
arity k generates two kinds of atomic formul: { R?+(e), an armative atom, states that the predicate R does hold on e. { R (e), a refutative atom, states that the predicate R does not hold on e. Notation R? (e) is used to denote either R+ (e) or R? (e). Now, negation can be seen as a de nable operation. :R+ (e) = R? (e) :R? (e) = R+ (e) :( ^ ) = (: ) _ (: ) :tt = ff :( _ ) = (: ) ^ (: ) :ff = tt It follows easily from the de nition that :(:') ', where ' means the syntactic equality of formul. Thus, negation is an involution. However, the structural and logical rules in Table 1 do not capture the idea that :' is indeed a negation of '. One way of enforcing this is to accept the following axioms. (`:) ` a; :a (:`) a; :a `
The rst says that assuming a and :a leads to contradiction; the second asserts that either a or :a always holds. It follows by induction that '; :' ` and ` '; :' hold for all '. The implication connective is also meta-de nable: ' ) =^ :' _ .
2.3 Equality Equality predicate plays a very special role in mathematics and in computer science. From the perspective of the intended embedding it presence is, simply, indispensable. Our axiomatization is equivalent to the well-established tradition, cf. [15, 8]. (=) ` e = e (=s) e = e0 ; '[e=x] ` '[e0 =x] The rst axiom schema asserts transitivity of equality. The second axiom schema relates substitution to equality. It captures the idea that equals may be substituted for equals. Formally, (=s) says that under assumption that terms e and e0 are equal one can replace some occurrences of e in a formula '[e=x] by e0 . Here, '[e=x] denotes the formula obtained as a result of syntactic substitution of e for all (free) occurrences of x in '. Thus, just like negation, substitution is presented as a part of the metatheory of LP= .
2.4 Theories
The logic of predicates with equality, i.e., its consequence relation, is generated by the set K of rules together with axiom schemas (`:), (:`), (=) and (=s). Any consequence relation which is obtained in such a way is called a theory. In case of classical logic an equivalent de nition is obtained by saying that a theory is determined by xing a proof system like K and a set of formul, viz. the `axioms'
of the theory. The reason is that, in the presence of implication, validity of any sequent is equivalent to validity of a sequent of the form ` . However, as we shall see in section 3, not all logics allow such simpli cation. That is why a more general notion of an axiom is accepted in this paper. Axioms are used also in many other situations. Typically, they are a convenient way of formalizing properties of the domain of objects under consideration.
3 Logic of Predicates with Explicit Substitutions
Substitution is normally considered, just like in section 2, as a part of the metatheory of a logic. This applies not only to logics, but to -calcul and type theories as well. It has been recently realized that more ecient implementations of functional languages can be achieved if one better controls the process of performing a substitution. This calls for frameworks with substitution as a primitive operation. Indeed, a variety of -calcul with explicit substitutions have already been considered, cf [1, 14]. All of them are 2-sorted | the old syntactic class of terms is retained while a new class of substitutions is added. The logic of predicates has already two sorts: the sort of terms and, built over terms, the sort of formul. So far nobody has considered adding explicit substitutions to it via a new syntactic sort. All attempts known to the author use the idea that substitutions behave as modal operators, see e.g., [16]. We have good reasons to consider substitutions as a new kind of atomic formul, cf. [5]. Predicates are eternal. They represent facts the truth of which does not depend on the context, or state, in which their truth is evaluated. Therefore, we call them Platonic here. Substitutions provide a formalization of the idea of an action or change of state in logic. Hence, we call them dynamic atoms. Since a substitution is an atom, a new logical connective is needed to express (pending) application of to, say, a predicate a. Let us use the tensor symbol
to denote the postulated connective. Thus, the above situation could be written as a. Consequently, assuming is associative, (0 a) a` ( 0 ) a. Hence, 0 would corresponds to composition of substitutions. Now, the question is this: Does such a logical connective exist? It turns out that we can view as a non-commutative multiplicative conjunction. The identity substitutions xx , for all x, should be neutral elements of . It is therefore natural to identify them all with I | the multiplicative truth constant. Since I captures the idea that \nothing changes" it is a good speci cation for a \do nothing" program, cf. [5]. Altogether, we are led to discover that the logic of predicates with explicit substitution, called LP= , is a fragment of intuitionistic non-commutative linear logic. Formul of LP= are given by the following grammar. A ::= j a j A A j I j A & A j > j A A j ? Unlike Girard we use ? instead of 0 to denote the neutral element of . This is to stress that it is the least element w.r.t. the derivability relation, and a dual to >. 3.1 A sequent system for LP= A sequent-style presentation of the logic is given in Table 2 in the appendix.
With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [10, 11, 12], to the non-commutative case, cf [7]. The exceptional axiom is (?). Its expected generalisation is ?; ?; ` A, as in [7]. However, the stronger axiom is not valid in our intended interpretation in quasi quantales as described in section 4 and in. [4]. Embedding the usual logic into LP= gives a good reason for not assuming that having falsehood as one of the assumptions always logically implies anything.
3.2 Platonic Formul
The idea underlying our embedding is that the logic of predicates lives as a Platonic sublanguage of LP= . More precisely, a formula is called Platonic if it contains neither explicit substitutions, nor nor I . To put the same statement in positive form, a Platonic formula is built from Platonic atoms and from additive conjunctions and disjunctions. Clearly, the Platonic formul are in 1{1 correspondence with formul of the predicate logic considered in section 2. Thus, with a slight abuse of notation, we let ' range over Platonic foruml of LP= .
3.3 Platonic axioms
Bear in mind that in a classical sequent ` sequences and stand for the additive conjunctions and disjunction of formul in and , respectively. Thus, axioms (:`) and (`:) responsible for negation correspond to (&:) R? (e) & :R? (e) ` ? and (:) > ` R? (e) :R? (e)
The linear counterpart of the re exivity of equality axiom schema is the following. (=) > ` e = e The other facet of equality, that equals can be substituted for equals, re ects the dynamic nature of substitution. This is treated in subsection 3.5. In rst order logic the conjunction distributes over disjunction, i.e., (A B ) & C a` (A & C ) (B & C ) and A (B & C ) a` (A B ) & (A C ): In linear logic not all of the above follow from the rules of Table 2. Consequently, we add those missing as axioms. (&) A (B & C ) a (A B ) & (A C ) (&) (A B ) & C ` (A & C ) (B & C )
3.4 Dynamic atoms
Substitutions, ranged over by , have the form xe .
3.5 Dynamic axioms
The additive connectives and predicate atoms are the Platonic ingredients of LP= . The dynamic ingredients are: the substitutions, multiplicative conjunction, and multiplicative truth. The dynamic features of the logic are captured by the following set of axioms. We add them all to the theory we build.
Substitution versus equality The principal property that equals may be substituted for equals can now be expressed as the following axiom. (=) e1 = e2 & ex1 ` ex2 Quick comparison with its predecessor in the logic of predicates reveals that the formula ' which played a dummy there is now, simply, removed. Pending substitutions Idea: B represents pending upon B.
Substitution pending on a Platonic atom is explained via meta-substitution. (R? ) e ? x R (e1 ; : : : ; ek ) a` R? (e1 [e=x] ; : : : ; ek [e=x]) One substitution pending upon another corresponds to their composition. Thus, composing two substitutions which concern the same variable results in aggregating the e ects of both in a single substitution. Just as in (R? ) pending explicit substitution in the world of formul is reduced/explained by metalevel substitution in the world of terms. All identity substitutions are equal I . (xx) (I ) x e1 e2 a` e2 [e1 =x] x a` I
x
x
x
Since simultaneous substitutions are not allowed, all we can do to explain the e ect of composition of two substitutions for di erent variables is to say how they commute. Again, the equality predicate is needed to state the axioms. (xz ) c1 = e2 [e1 =x] & e1 = c2 [c1 =z ] & ex1 ez2 ` cz1 cx2
Axiom (xz ) has the side-condition that x and z are di erent variables. The meaning of B in other cases is guided by distributivity axioms in subsection 3.5.
Platonic formul are facts The meaning of A B where A is a Platonic formul has not been described yet. Predicates, i.e., the Platonic atoms, were described as facts, i.e., formul the truth of which does not depend on the state. The same property holds for ?, and is enforced on the additive truth. The following axiom schemas capture the idea. (R? ) R? (e ; : : : ; e ) ` R?(e ; : : : ; e ) ? (> ) > ` > ? 1
k
1
k
Notice that axiom (> ) does not co-exist with the more general axiom ?; ?; ` A for the additive false. Together they give inconsistency: > ` ?. As a consequence of the above axioms one obtains the following result. Proposition 2. Platonic 's are a constant predicate transformers: a` B. The above proposition provides a technical justi cation of the idea that the Platonic formul do not depend on the state.
Distributivity axioms In the fragment of linear logic described in Table 2 the only distributivity law which is guaranteed is that of over binary additive disjunction . In general, as exempli ed by (R? ) and (> ), does not distribute over nullary additive disjunction ? on the right. But it does for dynamic atoms. (?) ? ` ? Let us recall that the right-sided distributivity ? A ` ? always holds by (?). Distributivity of over additive conjunctions is not guaranteed in general.
But we want it at least for the binary &. ( &) A (B & C ) a (A B ) & (A C ) (& ) (A & B ) C a (A C ) & (B C ) For nullary conjunction we impose left-sided distributivity only for dynamic atoms. (>) > a >
Its general form is not valid, cf. prop. 2. Its right-sided form always holds by (> ). Elimination of substitutions The consequences converse to those above follow from the logical rules, so the above give a method for elimination of substitutions and other non-Platonic ingredients. Consider the following inductive de nition which shows how it can be done when a formula has a form A '.
Lemma 3. For any Platonic ' and any A the formula dA 'e is a well de ned Platonic formula such that A ' a` dA 'e.
4 Translation of Logic of Predicates to LP= The general idea underlying our translation should be clear by now: Logic of predicates lives as the Platonic fragment of a LP= . Formally, given a classical formula ' its translation into LP= , denoted b'c, is de ned by induction on the structure of ' as follows. bR?(e1 ; : : : ; ek )c = R?(e1 ; : : : ; ek ) bttc = > b' ^ c = b'c & b c bff c = ? b' _ c = b'c b c Now, one can show that the function that eliminates tensor de ned in subsection 3.5 captures the performing of substitution. Lemma 4. d b'ce = b'c. Hence, b'c a` b'c in LP= , by Lemma 3. In LP the following holds
` i V
W
^
`
_
(9)
where and denote the conjunction and disjunction of all formul in and , respectively.
Accordingly, let bc denote the sequence obtained from by applying the translation elementwise. Keeping in mind the connection between the classical connectives and the linear connectives, and the equivalence (9), one de nes the translation on the level of sequents as follows.
&
b ` c def = bc `
M
b c:
4.1 Soundness of the translation
The reader should notice that the axioms responsible for negation adopted in LP= are translations of the negation axioms in predicate logic. Same applies to re exivity of equality axiom. This generalises to arbitrary rst order theories since every axiom of predicate logic corresponds to a Platonic axiom of LP= via translation. Somewhat more subtle situation is in the case of the axiom (=s). Its translation is e=e0 & b'[e=x]c ` b'[e0 =x]c. From prop. 2 it follows e=e0 ` e=e0 '. From h that i lemma 4 it follows that b'[e=x]c ` xe b'c and ex b'c ` b'[e0 =x]c. Thus, the task substantially simpli es. Now, it can be accomplished by the following proof. by (=) h i e=e & xe ` ex b'c ` b'c h i e ex b'c e = e & ; b ' c ` x by (&
) ? distributivity h i (e=e b'c) & ( xe b'c) ` (e=e & xe ) b'c (e=e & xe ) b'c ` ex b'c 0
0
0
0
0
0
h
0
0
0
i
(e=e b'c) & ( xe b'c) ` ex b'c 0
0
&
Before we formulate the main result of this section let usLintroduce a bit of notation. Given a set A of LP= axioms let bAc = f bc ` b c j ` 2 Ag denote its translation to LP= . With the above notation we can formulate the following result. Theorem 5. Let ` be a LP theory obtained by extending K with negation and equality axioms, and an arbitrary set of axioms A. Suppose that ` is a LP= theory obtained by extending L and such that the corresponding negation and equality axioms are satis ed together with (&), (&) and (& ) distributivity laws. Finally, suppose that ` admits all axioms in bAc. Then
` implies
&bc `
4.2 Conservativity of the translation
M
b c:
The result converse to Theorem 5 also holds. The proof presented in this section is based on semantical considerations. For the remaining part of this section let ` be a LP theory obtained by extending K with negation and equality axioms, and an arbitrary set of axioms A. Let ` be obtained by extending L with all axioms listed in section 3, i.e., negation, equality, substitutivity, distributivity and, nally, all axioms from the set bAc.
Denotational semantics of LP The notion of a model M for LP theory is given as follows. First, a set D is chosen, and for each function symbol f of arity k a function fM : Dk ! D called f 's interpretation. These data induce expression interpretation function EM [ ] : Exp ! (Var ! D) ! D, where Var and Exp are the syntactic classes of variables and expressions, respectively. Call Val =^ Var ! D the set of valuations of variables in D. For v : Val, x : Var and d : D de ne a new valuation, denoted v[x 7! d], by v[x 7! d]y =^ d whenever x and y are the same variables, and v[x 7! d]y =^ vy otherwise. It is assumed here that, given a particular vocabulary for building expressions, the reader is capable of lling in the details required to give the de nition of EM [ ] . The denotation of the formul of the logic of predicates in M is given by function FM [ ] : Pred ! Pow(Val)
where Pred is the set of LP= formul. Thus, each formula is identi ed with a set of valuations, viz., the valuations that satisfy the formula. This denotation function is determined once an interpretation RM Dk for each k-ary predicate symbol is given. It is assumed, that the interpretation =M D D is the diagonal binary relation on D. Let e =^ e1 ; : : : ; ek . FM [ R?+ (e)]] = fv : Val j (EM [ e1 ] v; : : : ; EM [ ek ] v) 2 RM g FM [ ff ] ) = ; FM [ R (e)]] = fv : Val j (EM [ e1 ] v; : : : ; EM [ ek ] v) 62 RM g FM [ tt] ) = Val FM [ ' _ ] = FM [ '] [ FM [ ] FM [ ' ^ ] = FM [ '] \ FM [ ] We may consider the substitution as an operation ondenotations of formul. Formally, given a set S of valuations and a substitution xe we de ne
e S =^ fv : Val j v[x 7! E [ e] v] 2 S g : x
h i
With the above notation one obtains the Substitution Lemma. Lemma 6 Substitution. FM [ '[e=x]]] = xe FM [ '] : Every model M determines a consequence relation j=M given by
j=M i
\
[
FM [ ] FM [ ] Call M a model of ` if ` j=M , i.e., whenever ` implies j=M . Let M be the set of all models of ` . Then Godel completeness result follows. Theorem 7 Completeness. ` i j=M , for all M 2 M.
Denotational semantics of LP= The notion of a model for a LP= theory=is
obtained by a trivial adaptation of the idea of LP-model. That is, each LP model is characterised by exactly the same set of ingredients as before. We take the same denotations for expressions. All the di erence are focused on the area of denotations of formul. Whereas before these were set of valuations, now they are functions between valuations.
Formally, the denotation of the formula of the logic of predicates with explicit substitutions in model M is given by function SM [ ] : Pred ! (Pow(Val) ! Pow(Val)) where Pred is the set of LP= formul. SM [ R? (e)]]S = FM [ R? (e)]] SM [ xe ] S = xe S SM [ A B ] S = SM [ A] (SM [ B ] S ) SM [ I ] S = S SM [ A B ] S = SM [ A] S [ SM [ B ] S SM [ ?] )S = ; SM [ A & B ] S = SM [ A] S \ SM [ B ] S SM [ >] )S = Val The semantic function extends to sequences in accord to the intuition that the meaning of a sequence stands for iterated tensor. Let us write ? j=M A i SM [ ? ] S SM [ A] S , for all S Val.
Proposition 8. 1. For any LP formula ' its dynamic meaning (via translation) is a constant function given by: SM [ b'c] S = FM [ '] . 2. b'c j=M b c implies ' j=M . It is time to come back to the issue of negation as a `second class citizen' in the logic of predicates. Taking it to the rst class league' would result in many problems. Simply, there can be no negation in our intended dynamic denotational semantics. At the moment we have not de ned how to interpret non-intuitionistic sequents ? ` , where is not a singleton. But, assuming that we have done it, somehow, the problem remains. Negation :A of a linear formula A should satisfy at least :A; A ` ?. But then :I; I ` ?, and hance :I a` ?. Since :> a` ? should also hold the negation could not be involution.
The result First, notice that under assumptions about ` and `, every model of the former is also a model of the latter. Proposition 9. Let M be a model of `. Then it is also a model of `. And now we are ready to formulate the main result.
Theorem 10. ` is a conservative extension of `
&
L
Proof. Let bc ` b c. Without loss of generality we may assume that and consist of a single formul ' and , respectively. Let M be a model of ` . By prop. 9 it is also a model of ` . So, b'c j=M b c. Hence, by prop. 8 one obtains ' j=M . Finally, ' ` , by prop. 7. ut
Conclusions We have shown that it is possible to give an explanation of the connections between the usual logic and linear logics based on the idea that the former is a platonic fragment of a speci c linear theory LP= | the logic of predicates with explicit substitutions.
Admittedly, the target linear logic is rather weak: it is non-commutative, has only the additives and multiplicative conjunctions, and even lacks one of the implications | see [4] for details. The interest in studying LP= stems from the fact that its formul are natural speci cations of imperative programs in a mechanised variant of Hoare logic. Thus, I is a natural and best speci cation for skip. Similarly, xe captures the meaning of the assignment x := e. Finally, sequential composition of programs is mimiced by tensor of their speci cations. See [5] for details.
Acknowledgements
I would like to acknowledge stimulating discussions with colleagues at ICS PAS, and in particular I would like to thank Beata Konikowska and Andrzej Tarlecki. The research presented here was partially sponsored by The State Committee for Scienti c Research, Grant No 2P301 007 04.
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Appendix ` ; ' '; ` '`' ; ` ; ` ; ; '; '; ` (` Contr) (Contr `) '; ` ` ; ` ` (` Weak) (Weak `) '; ` ` ; ; '; ; ` ` ; '; ; (Exch `) (` Exch) ; ; '; ` ` ; ; '; (` tt) (ff `) ff ; ` ` ; tt '; ` ; ` ` ; ' ` ; (_`) (`^) ' _ ; ` ` ; ' ^ ` ; ' '; ` (`_ l) (^` l) ' ^ ; ` ` ; ' _ ` ; ; ` (`_ r) (^` r) ' ^ ; ` ` ; ' _ = Table 1. A sequent system K for PL . (Id)
0
0
0
0
0
0
0
A; ` B (Cut) ? ` A; ?; ; `B 0
(Re ) A ` A (LI ) ?;?;I;``AA
0
(L ) ?;?;AA; B;B;``CC A; ` C (L&-L) ?; ?; A & B; ` C
B; ` C (L&-R) ?; ?; A & B; ` C (?) ?; ? ` A A (R-L) ? ?` A` B
0
(Cut)
(RI ) ` I A `B (R ) ? ?;` ` A B (>) ? ` > (R&) ? `?A` A &?B` B
(L) ?; A; ?;`AC B; ?;`B;C ` C
B (R-R) ? ?` A` B Table 2. A sequent system L for NILL.
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